On Spherical Designs of Some Harmonic Indices

A finite subset Y on the unit sphere Sn−1 ⊆ Rn is called a spherical design of harmonic index t, if the following condition is satisfied: ∑ x∈Y f(x) = 0 for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree t. Also, for a subset T of N = {1, 2, · · · }, a finite subset Y ⊆ Sn−1 is called a spherical design of harmonic index T, if ∑ x∈Y f(x) = 0 is satisfied for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree k with k ∈ T . In the present paper we first study Fisher type lower bounds for the sizes of spherical designs of harmonic index t (or for harmonic index T ). We also study ‘tight’ spherical designs of harmonic index t or index T . Here ‘tight’ means that the size of Y attains the lower bound for this Fisher type inequality. The classification problem of tight spherical designs of harmonic index t was started by Bannai-OkudaTagami (2015), and the case t = 4 was completed by Okuda-Yu (2016). In this paper we show the classification (non-existence) of tight spherical designs of harmonic index 6 and 8, as well as the asymptotic non-existence of tight spherical designs of harmonic index 2e for general e > 3. We also study the existence problem for tight spherical designs of harmonic index T for some T , in particular, including index T = {8, 4}.